No, the Church is not opposed to capital punishment under all circumstances. What it teaches is since today we have prisons, the instances in which capital punishment may be necessary are rare. (Catechism of the Catholic Church, 2267).
No, the Church does not deny the possibility of a just war. What it teaches is that countries should wage war reluctantly, and only for justice. (Catechism of the Catholic Church, 2309)
No, the Church is not socialist. What it teaches is that the institution of private property is justified because of its contribution to the common good. (Catechism of the Catholic Church, 2403.)
Here is a test for Catholic social activists: How well do you know, and how seriously do you take, the principle of subsidiarity? Most haven’t heard of it.
“Just as it is gravely wrong to take from individuals what they can accomplish by their own initiative and industry and give it to the community, so also it is an injustice and at the same time a grave evil and disturbance of right order to assign to a greater and higher association what lesser and subordinate organizations can do. For every social activity ought of its very nature to furnish help to the members of the body social, and never destroy and absorb them.” (Pius XI, Quadragesimo Anno)